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64

Chapter 9 Project: Get the Picture

Beginning the Chapter ProjectWhen a book is being made, artists, designers, and photographers work with writers and editors to make the pages visually attractive. Th ese professionals often work with patterns involving arithmetic and geometric sequences.

In this project, you will see how perspective aff ects perceived lengths and distances. You will use grids to change the sizes of drawings. You also will learn how a designer crops a photo, then enlarges or reduces it.

ActivitiesActivity 1: ResearchingResearch the concepts of one- and two-point perspective and vanishing points in art.

• Measure the lengths of the arrows shown at the right. What is the relationship between these lengths? How does this relate to your research on perspective?

• Trace the four arrows at the right, moving the paper to the left after tracing the longest arrow so that it is further away from the others than it is now. What do you notice?

• Make a simple drawing of three or more similar objects whose lengths can be represented by an arithmetic sequence. Write the corresponding arithmetic sequence, and a recursive or explicit formula for that sequence.

Activity 2: DesigningWhen a book is made, a designer or artist may change the size of an original sketch to fi t the space available on a page. One way to change the dimensions of a sketch is to use graph paper with diff erent size squares.

• Draw a fi gure or design on a sheet of graph paper. Label this Figure 1 and record its approximate dimensions.

• Enlarge the original fi gure by copying each portion of Figure 1, square by square, onto larger squares. Label this Figure 2 and record its dimensions.

• Use a ratio to compare the dimensions of Figure 1 to the dimensions of Figure 2. If the same ratio is used to enlarge Figure 2, what would the dimensions of the new fi gure be? Draw this fi gure, label it Figure 3, and record its dimensions.

• Explain why the lengths of the three fi gures form a geometric sequence.• Write a geometric sequence corresponding to these lengths, and a recursive or explicit

formula for that sequence.

Figure 1 Figure 2

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65

Chapter 9 Project: Get the Picture (continued)

Activity 3: AnalyzingPhotographs are often cropped so that only part of the photograph remains. Th en, this cropped portion can be reduced or enlarged. Choose a photograph in a textbook. Place a piece of paper over the photograph, trace its original size, and draw a rectangle to indicate a portion of the photograph that you would like to crop. Draw a diagonal from the lower left corner to the upper right corner of the rectangular cropped area. If this diagonal is extended through the upper right corner of the cropped area, and a point selected anywhere along the diagonal or its extension, then the rectangle having the chosen point as its upper right corner (and the same lower left corner as the original cropped area) will have dimensions that are proportional to the dimensions of the cropped area.

• Measure the dimensions and the length of the diagonal of the cropped area.

• Write the fi rst four terms of an arithmetic sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?

• Write the fi rst four terms of an geometric sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?

Finishing the ProjectTh e answers to the activities should help you complete your project. Prepare a presentation or demonstration that summarizes how an artist, a designer, or a photographer uses sequences. Present this information to your classmates. Th en discuss the sequences you made.

Refl ect and ReviseReview your summary. Are your drawings clear and correct? Are your sequences accurate? Practice your presentation in front of at least two people before presenting it to the class. Ask for their suggestions for improvement.

Extending the ProjectGeometric and arithmetic patterns are used in other aspects of design and in other careers. Research other areas where sequences are applied.